AGE STRUCTURED MODELS AARON A. KING AND HELEN J. WEARING Licensed under the Creative Commons attribution-noncommercial license http://creativecommons.org/licenses/by-nc/3.0/. Please share & remix noncommercially, mentioning its origin. Contents 1. Beyond homogeneous populations: age structure 2 2. Getting more realistic: nobody's youth is exponential 8 3. What do real contact networks look like? 11 References 12 Date: June 24, 2011. 1 2 KING AND WEARING 1. Beyond homogeneous populations: age structure α B Sj Sa λJ λA α Ij Ia γ γ α Rj Ra D Figure 1. SIR dynamics in an age-structured population. Today's first lecture showed how force of infection can vary with age. What sort of mechanisms might give rise to these effects? Here we'll see to what extent we can infer these mechanisms on the basis of age- specific incidence and seroprevalence data. We'll start by introducing age into the simplest mechanistic AGE STRUCTURED MODELS 3 model we can think of, which has separate classes for juveniles and adults: dS J = −λ S dt J J dS A = −λ S dt A A dIJ = λJ SJ − γ IJ dt (1) dI A = λ S − γ I dt A A A dR J = + γ I dt J dR A = + γ I dt A The λs denote the age-specific force of infections: λJ = βJJ IJ + βJA IA (2) λA = βAJ IJ + βAA IA In this model, each population can infect each other but the infection moves through the populations separately. Let's simulate such a model. To make things concrete, we'll assume that the transmission rates β are greater within groups than between them. b1 <- 0.005 b2 <- 0.005 gamma <- 10 ja.model <- function (t, x, ...) { s <- x[c("Sj","Sa")] # susceptibles i <- x[c("Ij","Ia")] # infecteds r <- x[c("Rj","Ra")] # recovereds n <- s+i+r # total pop lambda.j <- (b1+b2)*i[1]+b1*i[2] # juv. force of infection lambda.a <- b1*i[1]+(b1+b2)*i[2] # adult. force of infection list( c( -lambda.j*s[1], -lambda.a*s[2], lambda.j*s[1]-gamma*i[1], lambda.a*s[2]-gamma*i[2], gamma*i[1], gamma*i[2] ) ) } 4 KING AND WEARING require(deSolve) ## initial conditions yinit <- c(Sj=2000,Sa=1000,Ij=0,Ia=1,Rj=0,Ra=0) sol <- ode( y=yinit, times=seq(0,2,by=0.01), func=ja.model ) plot(sol) dim(sol) head(sol) plot(sol,log='y') time <- sol[,1] # time y <- sol[,-1] # all other variables n <- apply(y,1,sum) # population size prop <- y/n # fractions subsampled.prop <- prop[seq(1,length(time),by=10),] subsampled.time <- time[seq(1,length(time),by=10)] barplot( t(subsampled.prop), names.arg=subsampled.time, xlab='time',main='Population structure', space=0, col=c( rgb(0.5,1,0.5), rgb(0,1,0), rgb(1,0.5,0.5), rgb(1,0,0), rgb(0.5,0.5,1), rgb(0,0,1) ), legend = colnames(prop), args.legend=list(bg="white") ) AGE STRUCTURED MODELS 5 Population structure 1.0 Ra Rj Ia 0.8 Ij Sa Sj 0.6 0.4 0.2 0.0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time Figure 2. The population structure changes over the course of the epidemic. The results of the above are plotted in Fig. 2. However, we haven't yet modeled the aging process. We can do this very simply using the same ingredients that go into the basic SIR model. In that model, the waiting times in the S and I classes are exponential. Let's assume the same thing about the aging process. We'll also add in births. 6 KING AND WEARING dS J = −λ S + B − α S dt J J J dS A = −λ S − µ S + α S dt A A A J dIJ = λJ SJ − γ IJ − α IJ dt (3) dI A = λ S − γ I − µ I + α I dt A A A A J dR J = + γ I − α R dt J J dR A = + γ I − µ R + α R dt A A J Now, let's simulate this model, under the same assumptions about transmission rates as above. b1 <- 0.005 b2 <- 0.005 gamma <- 10 births <- 100 da <- c(20,60) # alpha = 1/da ja.demog.model <- function (t, x, ...) { s <- x[c("Sj","Sa")] # susceptibles i <- x[c("Ij","Ia")] # infecteds r <- x[c("Rj","Ra")] # recovereds n <- s+i+r # total pop lambda.j <- (b1+b2)*i[1]+b1*i[2] # juv. force of infection lambda.a <- b1*i[1]+(b1+b2)*i[2] # adult. force of infection alpha <- 1/da list( c( -lambda.j*s[1] -alpha[1]*s[1]+births, -lambda.a*s[2] +alpha[1]*s[1]-alpha[2]*s[2], lambda.j*s[1]-gamma*i[1]-alpha[1]*i[1], lambda.a*s[2]-gamma*i[2]+alpha[1]*i[1]-alpha[2]*i[2], gamma*i[1]-alpha[1]*r[1], gamma*i[2]+alpha[1]*r[1]-alpha[2]*r[2] ) ) } Note that in this function, µ =alpha[2], i.e., death is just another age class. require(deSolve) ## initial conditions yinit <- c(Sj=2000,Sa=1000,Ij=0,Ia=1,Rj=0,Ra=5000) sol <- ode( y=yinit, times=seq(0,200,by=0.1), func=ja.demog.model ) AGE STRUCTURED MODELS 7 plot(sol) dim(sol) head(sol) plot(sol,log='y') equil <- drop(tail(sol,1))[-1] n <- equil[c("Sj","Sa")]+equil[c("Ij","Ia")]+equil[c("Rj","Ra")] seroprev <- equil[c("Rj","Ra")]/n names(seroprev) <- c("J","A") barplot(height=seroprev,width=da,ylab="seroprevalence") To compute R0, we need to know the stable age distribution of the population, which we can find by ∗ ∗ solving for the disease-free equilibrium: SJ = B/α and SA = B/µ. Now we have the stable age distribution, we can calculate R0 by constructing the next generation matrix. Intuitively, this is a matrix that specifies how many new age-specific infections are generated by a typical infected individual of each age class (in a fully susceptible population). For example, let's consider an infected adult and ask how many new juvenile infections it generates: this is the product of the number of susceptible juveniles (from the stable age distribution), the per capita transmission rate from adults ∗ to juveniles and the average duration of infection, i.e. SJ × βJA × 1=(γ + µ). This forms one element of our next generation matrix. The other elements look very similar, except there are extra terms when we consider an infected juvenile because there is a (very small) chance they may age during the infectious period and therefore cause new infections as an adult: ∗ ∗ ∗ SJ βJJ α SJ βJA SJ βJA ! (γ+α) + (γ+µ) (γ+µ) (γ+µ) NGM = ∗ ∗ ∗ (4) SAβAJ α SAβAA SAβAA (γ+α) + (γ+µ) (γ+µ) (γ+µ) R0 can then be computed as the dominant eigenvalue (i.e., the one with the largest real part) of this matrix. In R, we do alpha <- 1/da[1] mu <- 1/da[2] n <- births/c(alpha,mu) beta <- matrix(c(b1+b2,b1,b1,b1+b2),nrow=2,ncol=2) ngm <- matrix( c( n[1]*beta[1,1]/(gamma+alpha)+alpha/(gamma+mu)*n[1]*beta[1,2]/(gamma+mu), n[2]*beta[2,1]/(gamma+alpha)+alpha/(gamma+mu)*n[2]*beta[2,2]/(gamma+mu), n[1]*beta[1,2]/(gamma+mu), n[2]*beta[2,2]/(gamma+mu) ), nrow=2, ncol=2 ) eigen(ngm) eigen(ngm,only.values=TRUE) max(Re(eigen(ngm,only.values=T)$values)) 8 KING AND WEARING 2. Getting more realistic: nobody's youth is exponential In the model above, the aging process follows an exponential distribution, which means that whether an individual is 1 year old or 10 years old, the chance of them becoming an adult is the same! To improve on this, we can assume that the time a juvenile must wait before becoming an adult follows a gamma distribution. This is equivalent to saying that the waiting time is a sum of some number of exponential distributions. This suggests that we can achieve such a distribution by adding age classes to the model, so that becoming an adult means passing through some number of stages. We'll use 30 age classes, and since they don't have to be of equal duration, we'll assume that they're not. Specifically, we'll have 20 1-yr age classes to take us up to adulthood and break adults into 10 age classes of 5 yr duration each. The last one will take us up through age 80. Now, when we had just two age classes, we could write out each of the equations easily enough, but now that we're going to have 30, we'll need to be more systematic. In particular, we'll need to think of β as a matrix of transmission rates. Let's see how to define such a matrix in R. So that we don't change too many things all at once, let's keep the same contact structure as in the juvenile-adult model. ages <- c(seq(1,20,by=1),seq(25,65,by=5),80) # upper end of age classes da <- diff(c(0,ages)) # widths of age classes beta <- matrix(nrow=30,ncol=30) beta[1:20,1:20] <- b1+b2 beta[21:30,21:30] <- b1+b2 beta[1:20,21:30] <- b1 beta[21:30,1:20] <- b1 dim(beta) filled.contour(beta,plot.title=title(main="WAIFW matrix")) Let's use the techniques we learned from John to simulate the re-introduction of a pathogen into a population of hosts.